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Mirrors > Home > ILE Home > Th. List > xpdisj1 | Unicode version |
Description: Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
Ref | Expression |
---|---|
xpdisj1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inxp 4722 | . 2 | |
2 | xpeq1 4602 | . . 3 | |
3 | 0xp 4668 | . . 3 | |
4 | 2, 3 | eqtrdi 2206 | . 2 |
5 | 1, 4 | syl5eq 2202 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 cin 3101 c0 3395 cxp 4586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-opab 4028 df-xp 4594 df-rel 4595 |
This theorem is referenced by: djudisj 5015 xp01disjl 6383 xpfi 6876 djuinr 7009 |
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